I am trying to use the Rayleigh-Ritz method to calculate an approximate solution to the extremum problem with the following functional: $$ L[y]=\int\int_D (u_x^2+u_y^2+u^2-2xyu)\,dx\,dy, $$ $D$ is the unit square i.e. $0 \leq x \leq 1, 0 \leq y \leq 1.$ Also $u=0$ on the boundary of $D$.
I have chosen to use the trial function: $$ \phi(x,y)=cxy(1-x)(1-y) $$ Where $c$ is a constant that I need to find.
I am familiar with using the Rayleigh-Ritz method most of the time, however this question I am not sure of. Is it possible to convert the problem to a Sturm-Liouville ration type?
Thanks for your help.